Sums of Powers of the Natural Numbers

Extract

When one first comes across the formulae

one is immediately struck by the fact that the second sum is the square of the first. The usual methods of proof give no reason to expect that the sums will be related in such a pleasant and easily memorable way; one is left with the impression that this is a happy accident. The following proof may give some entertainment because it throws light on this point; it emerged unexpectedly out of the tortuous process of making up a question for a recent Scholarship examination.